Remove the smart host from the SMTP Virtual server - it should not be configured there.
Then configure an SMTP Connector instead to hold the smart host.
If the second smart host is using the same authentication properties, then you can put it on the same connector, separated by a comma.
If the second smart host is using different authentication properties then create a new SMTP connector, set the cost as 1 on the new connector as well and Exchange will load balance them.

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When you have a smart host set on the SMTP Virtual server, it forces all email to go over the smart host. That includes internal email when you are using multiple servers.
It is best practise to use an SMTP Connector instead.

There are many sites out there, including some from major vendors that state to use the Smart Host on the SMTP virtual server, but that will cause problems.

No.
Internal emails stay on the server.
The major issue comes in migrations, but using a smart host on the SMTP VS gives you no control over email at all. If you wanted to route email for a specific domain another way, you would be unable to, because the SMTP VS forces the email down the route to the smart host.